Is there a complete classification of compact surface $S\subset \mathbb{R}^3$ for which $\Delta \kappa \geq 0$ where $\kappa $ is the Gaussian curvature of $S$.

Does every (compact) $2$ dimensional manifold admit a Riemannian metric with this property?

Is there a name for this property in $2$ (or higher dimensions)?

| cite | improve this question | | | | |

The Laplacian of any function $f \colon S \to \mathbb{R}$ is $\Delta f$ defined by $\Delta f \, dA = -d(*df)$, so is exact, and hence has integral zero if $S$ has empty boundary. So if $\Delta f \ge 0$ then $0=\int \Delta f$ forces $\Delta f=0$ everywhere, and so (if $S$ is connected) $f$ is constant.

| cite | improve this answer | | | | |
  • 3
    $\begingroup$ This answer proves that $\kappa$ must be constant. To answer the original question it remains to say that yes, on every compact surface, there is a metric of constant curvature. The sign of this constant depends only on the topology of the surface: + for the sphere, 0 for tori, and - for every other surface. $\endgroup$ – Alexandre Eremenko Dec 25 '17 at 18:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.